Applications of Certain Conic Domains to a Subclass of <i>q</i>-Starlike Functions Associated with the Janowski Functions
In our present investigation, with the help of the basic (or <i>q</i>-) calculus, we first define a new domain which involves the Janowski function. We also define a new subclass of the class of <i>q</i>-starlike functions, which maps the open unit disk <inline-formula>...
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| Format: | Article |
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MDPI AG
2021-03-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/4/574 |
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| author | Bilal Khan Hari Mohan Srivastava Nazar Khan Maslina Darus Qazi Zahoor Ahmad Muhammad Tahir |
| author_facet | Bilal Khan Hari Mohan Srivastava Nazar Khan Maslina Darus Qazi Zahoor Ahmad Muhammad Tahir |
| author_sort | Bilal Khan |
| collection | DOAJ |
| description | In our present investigation, with the help of the basic (or <i>q</i>-) calculus, we first define a new domain which involves the Janowski function. We also define a new subclass of the class of <i>q</i>-starlike functions, which maps the open unit disk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">U</mi></semantics></math></inline-formula>, given by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">U</mi><mo>=</mo><mo> </mo><mfenced separators="" open="{" close="}"><mi>z</mi><mo>:</mo><mi>z</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo> </mo><mi>and</mi><mo> </mo><mfenced open="|" close="|"><mi>z</mi></mfenced><mo> </mo><mo><</mo><mn>1</mn></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> onto this generalized conic type domain. We study here some such potentially useful results as, for example, the sufficient conditions, closure results, the Fekete-Szegö type inequalities and distortion theorems. We also obtain the lower bounds for the ratio of some functions which belong to this newly-defined function class and for the sequences of the partial sums. Our results are shown to be connected with several earlier works related to the field of our present investigation. Finally, in the concluding section, we have chosen to reiterate the well-demonstrated fact that any attempt to produce the rather straightforward <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>-variations of the results, which we have presented in this article, will be a rather trivial and inconsequential exercise, simply because the additional parameter <i>p</i> is obviously redundant. |
| first_indexed | 2024-03-10T12:45:37Z |
| format | Article |
| id | doaj.art-7fd2be2907f04cd99b1f2549cd7f1614 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T12:45:37Z |
| publishDate | 2021-03-01 |
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| series | Symmetry |
| spelling | doaj.art-7fd2be2907f04cd99b1f2549cd7f16142023-11-21T13:30:44ZengMDPI AGSymmetry2073-89942021-03-0113457410.3390/sym13040574Applications of Certain Conic Domains to a Subclass of <i>q</i>-Starlike Functions Associated with the Janowski FunctionsBilal Khan0Hari Mohan Srivastava1Nazar Khan2Maslina Darus3Qazi Zahoor Ahmad4Muhammad Tahir5School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai 200241, ChinaDepartment of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, CanadaDepartment of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, PakistanDepartment of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor, MalaysiaGovernment Akhtar Nawaz Khan (Shaheed) Degree College KTS, Haripur 22620, PakistanDepartment of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, PakistanIn our present investigation, with the help of the basic (or <i>q</i>-) calculus, we first define a new domain which involves the Janowski function. We also define a new subclass of the class of <i>q</i>-starlike functions, which maps the open unit disk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">U</mi></semantics></math></inline-formula>, given by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">U</mi><mo>=</mo><mo> </mo><mfenced separators="" open="{" close="}"><mi>z</mi><mo>:</mo><mi>z</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo> </mo><mi>and</mi><mo> </mo><mfenced open="|" close="|"><mi>z</mi></mfenced><mo> </mo><mo><</mo><mn>1</mn></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> onto this generalized conic type domain. We study here some such potentially useful results as, for example, the sufficient conditions, closure results, the Fekete-Szegö type inequalities and distortion theorems. We also obtain the lower bounds for the ratio of some functions which belong to this newly-defined function class and for the sequences of the partial sums. Our results are shown to be connected with several earlier works related to the field of our present investigation. Finally, in the concluding section, we have chosen to reiterate the well-demonstrated fact that any attempt to produce the rather straightforward <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>-variations of the results, which we have presented in this article, will be a rather trivial and inconsequential exercise, simply because the additional parameter <i>p</i> is obviously redundant.https://www.mdpi.com/2073-8994/13/4/574analytic functionsconic domainsstarlike functions<i>k</i>-uniformly starlike functions<i>q</i>-starlike functionssufficient conditions |
| spellingShingle | Bilal Khan Hari Mohan Srivastava Nazar Khan Maslina Darus Qazi Zahoor Ahmad Muhammad Tahir Applications of Certain Conic Domains to a Subclass of <i>q</i>-Starlike Functions Associated with the Janowski Functions Symmetry analytic functions conic domains starlike functions <i>k</i>-uniformly starlike functions <i>q</i>-starlike functions sufficient conditions |
| title | Applications of Certain Conic Domains to a Subclass of <i>q</i>-Starlike Functions Associated with the Janowski Functions |
| title_full | Applications of Certain Conic Domains to a Subclass of <i>q</i>-Starlike Functions Associated with the Janowski Functions |
| title_fullStr | Applications of Certain Conic Domains to a Subclass of <i>q</i>-Starlike Functions Associated with the Janowski Functions |
| title_full_unstemmed | Applications of Certain Conic Domains to a Subclass of <i>q</i>-Starlike Functions Associated with the Janowski Functions |
| title_short | Applications of Certain Conic Domains to a Subclass of <i>q</i>-Starlike Functions Associated with the Janowski Functions |
| title_sort | applications of certain conic domains to a subclass of i q i starlike functions associated with the janowski functions |
| topic | analytic functions conic domains starlike functions <i>k</i>-uniformly starlike functions <i>q</i>-starlike functions sufficient conditions |
| url | https://www.mdpi.com/2073-8994/13/4/574 |
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